Your Brain on Math: 3/16/2014. Making Sense of Number Sense. Sara Stetson

Transcription

1 Your Brain on Math: Making Sense of Number Sense Sara Stetson NHTM Spring Conference March 17, 2014 Stetson, 2014 Agenda What is number sense? Core Systems of number: -Core system 1: Approximate Number System -Core System 2: Exact Number Object Attention System Space and number: Integer representation Your Brain on Math - Dyscalculia Recommendations for instruction Billion # Students Days of Counting 1.20 days 2 3 days 7 7 days 1 14 days 1 15 days days 1 31 days days days days days days 12 Years 19 Years 20 Years 27.4 Years Years 1

2 Infants What is Number Sense? Animals Indigenous People Brain studies The Approximate Number System Approximate Number System Fight or Flight Foraging vs. 2

3 Approximate Number System Number Sense Number sense and the Approximate Number System are the same thing = Approximate Number System Modality invariance Objects, sounds, actions (e.g., Brannon, 2001; Dehaene, et al., 2004; Lipton & Spelke, 2003; McCrink & Wynn, 2005; Wood & Spelke, 2005; Xu, 2002) 3

4 Approximate Number System No tracking of individual elements Gross estimation: Quantity comparison, addition, and subtraction (Brannon, 2001; Xu, 2002; Lipton & Spelke, 2003; Wood & Spelke, 2005; McCrink & Wynn, 2005; dehevia& Spelke, 2008) Approximate Number System Spontaneous linkage to space A natural propensity to represent quantity along a mental number line (dehevia& Spelke, 2008) Approximate Number System Ratio-dependence (e.g., Brannon, 2001; Berger, et al., 2006; Libertus & Brannon, 2009 Lipton & Spelke, 2003; McCrink & Wynn, 2005; Wood & Spelke, 2005) 4

5 Approximate Number System: Ratio Dependence 6 month-old babies Discriminate 8 vs. 16, But not 8 vs. 12, Ratio 1:2 9 month-old babies Discriminate 16 vs. 24 Ratio 2:3 Adults: ratio 9:10 What is Ratio-Dependence? (Dehaene, 2003; Gallistel & Gelman, 2000; Huntley-Fenner, 2001; Temple & Posner, 1998) Approximate Number System Precision Representation of a number Low variability vs. High variability Target Number (Neider, 2004; Neider & Miller, 2003) 5

6 Approximate Number System Precision (Halberda, in press) Weber s Law If the intensity of a sensation is doubled from an initial value and the sensation is noted, the intensity will have to be doubled again to give the same perception of increase in sensation X 10 X 100 Weber s Law and Number Discriminate 13 from 10 at 90% Numerical distance of 3 Double the reference array to 20: How far from this numerosity do we need to go to again reach 90% correct discrimination? We need to double the numerical distance to 6, presenting an array of 26 dots 6

7 Approximate Number System Precision Low Weber fraction= low variability and more precision High Weber fraction = high variability and less precision SD = n(w) Target Number (Halberda, in press; Neider, 2004; Neider & Miller, 2003) Approximate System:Universal (Pica, Izard, Lemer, & Dehaene, 2004) ANS and Math Gilmore et al. (2007) 7

8 The Exact Number System The Exact Number System Infant Addition and Subtraction Infant object permanence Infant number processing (Wynn, 1992) 8

9 Choice ratio Infant Addition and Subtraction Berger, et al 2006 Exact System: Limits Set Size Limit: 3-4 Explicit number or object files Numerical cognition or objectbased attention? Persistence Task Search Task (Feigenson& Carey, 2003; Feigenson, Carey & Hauser, 2002; VanMarle, 2005) Exact System: Limits 12 month olds- Manual Search 1 vs 4 2 vs 4 3 vs 4 2 vs 3 1 vs Percent choosing more (Adapted from Hauser, et al., 2000) 9

10 Exact System: Universal Set Size Limit: 3-4 Explicit number or object files Domain limit objects Numerical cognition or object-based attention? (Rugani, Regolin, & Vallortigara, 2010) The Linear Number Properties The log-to-linear shift is hard Why is Number Mapping Difficult? Requires a log-to-linear shift There are several number properties Most number properties are spatial but numbers are expressed symbolically Neither core system represents integers 10

11 Cardinality Principle Put one fish in the pond Put two fish in the pond 2 yrs. 5 yrs. (Butterworth, 2005; Carey & Sarnecka, 2006, 2008; Lipton & Spelke, 2006; Wynn, 1990) Count each item only once 1:1 Principle (Butterworth, 2006; Gelman & Gallistel, 1978; Wynn, 1990) Wishes and dreams Abstraction Principle (Butterworth, 2006; Gelman & Gallistel, 1978; Wynn, 1990) 11

12 Order Irrelevance Principle Start anywhere (Butterworth, 2006; Gelman & Gallistel, 1978; Wynn, 1990) Stable Order Principle Start at the beginning Go in order Mental Numberline (Butterworth, 2006; Gelman & Gallistel, 1978; Wynn, 1990) Successor Function How many vs. how much Infinity N + 1 N (Butterworth, 2006; Gelman & Gallistel, 1978; Wynn, 1990) 12

13 Number and Space Numbers and Space: Continuous Extent (Brannon, Lutz, & Cordes, 2006; Cordes & Brannon, 2009; De Hevia & Spelke, 2010) ANS and Space: Number vs. Continuous Extent 13

14 Numbers and Space: Magnitude Numbers and Space: Relative Magnitude Numbers and Space: Position (Cantlon & Brannon, 2006;Lyons & Beilock, 2011; Picozi, et al., 2010) 14

15 Integer Representation Approximate System - Approximate - Sets - Continuous extent - No limit - Multi-modal - Spatial = Mental numberline - No Lanugage Exact System - Exact - Individuals - Discrete objects - Limit Visual - Object-based = No numberline - Language (eg., 1, 2, 3) Measuring Number Sense How do We Measure Number Sense? Quantity Comparison (Halberda et al., 2008) 15

16 Quantity Comparison: Sample Output Quantity Comparison: Sample Output How do We Measure Number Sense? Number-to-Position (Halberda et al., 2008; Siegler & Booth, 2004) 16

17 Number-to-Position: Sample Output (Siegler & Booth, 2004) Dyscalculia Your Brain on Math Horizontal Intraparietal Sulcus Calculation Calculation/Language Grasping objects Manual tasks Visuo-spatial tasks Attention Saccades (rapid eye fixations) (Adapted from Dehaene, 2013) (Simon, Mangin, Le Bihan, Cohen, & Dehaene, 2002) 17

18 Your Brain on Math Using number sense activates the IPS Tasks that activate this region: Comparison of numbers Subtraction Approximation Thinking about space Non-Symbolic Tasks Brains on Math Neurotypical brain processes numerosities in Intraparietal Sulcus Castelli et al, 2006 Part of the calculation network Zago et al, 2001 Is structurally abnormal in dyscalculics Isaacs et al, 2001 Twin study shows Neural and behavioural abnormalities correlate Both are heritable (slide adapted from Butterworth, 2013) Ranpura et al., 2013 Brains on Math Why is numerical and spatial information related/confused? Monkeys taught to discriminate number and length (continuous extent) Neurons that encode number, length/space, and both 18

19 Dyscalculia Severe difficulty in mathematics, not explained by general cognitive difficulties or educational opportunities Prevalence: around 5-6% (same as dyslexia!) Fraction of the studies compared to dyslexia (Butterworth, 2002; Shalev & Gross-Tsur, 2001; Geary, 1993, 2004) Counting strategies (eg., counting on from larger vs. counting all) Prolonged use of finger counting (slow, inaccurate, finger agnosia) Decomposing numbers (e.g. recognizing that 10 is made up of 5 and 5) Place value Multi-step procedures Observable Deficits (Wilson, 2008) Cognitive Deficits Difficulty representing quantity (number sense). Large Weber fraction/poor quantity comparison Mental Numberline slow to develop/logarithmic Where is 7? 0 10 Slow to enumerate set of objects Slow to subitize sets of 1-4 objects Number symbols not automatic Number stroop task (Rouselle & Nöel, 2007)

20 Subtypes Number sense = Spatial Verbal Difficulty with counting, fact retrieval, word problems Associated with dyslexia? Executive Difficulty with fact retrieval, use of strategy/procedure Associated with ADHD? Wilson & Dehaene (2007) Instruction Weber Fraction and Math Performance Halberda et al., teenagers of 14 y.o. Assessment of the acuity of their number sense Record of their past school performance in math Quantity Comparison Halberda, Mazzocco & Feigenson (2008) 20

21 Weber Fraction and Math Performance Score at the math test at 8 y.o. (high = good) W: acuity of number sense, at 14 y.o., low = good Manipulatives: Continuous Magnitude A display of continuous magnitudes involved in trained addition problems promotes logto-linear mapping Babies can map approximate sets to length (Spelke) and so this tendency is strong Board Games: Continuous Magnitude Linear but not circular board game play encourages log-to-linear shifting Explicit feedback on number-line anchoring (eg., calibration priming) encourages log-to-linear mapping 21

22 Color/Number (Ramani & Siegler, 2008 Adaptive CAI Linking digits, sets and space Practicing comparison, addition and subtraction (Adapted from Wilson, 2008) (Wilson et al., 2006) Number Race 22

23 Faster in dots enumeration Increased accuracy in subtraction (but not addition) Increased speed in magnitude comparison, but accuracy difference non-significant... Number Sense Games Number Bonds Initially 3 secs to click on matching bond adjusts to performance Stages: length + colour, length + colour + digit, length + digit, digit (Butterworth & Laurillard) 23

24 Dyscalculia: Rescue Calcularis 16 control and 16 MLD 25 sessions, 15 minutes 5 weeks Improvement in calculation both groups (Kucian, et al., 2011) There s an App for That Numberline Tap My Numberline Numberline Frog Thank You sstetson@rivier.edu 24